This invention is concerned generally with measurement devices, and, more particularly, with transfer function measurement devices.
Transfer, impedance, and frequency responses are terms used in describing the gain and phase response of an interconnection of physical devices as a function of frequency. These measurements are most useful in describing linear systems, but they may also be used to describe a linear approximation, within a band of frequencies, to a nonlinear system. These transfer functions are conventionally measured by applying a sinusoidal stimulus at a single frequency to the system under test while observing the gain and phase relationships between the stimulus and the system response. This measurement method has certain limitations. If single frequency stimuli are applied successively, then the measurement at each frequency will take a time equal to several of the longest time constants in the measurement system or the system under test (SUT). If the SUT has any nonlinearities, then the distortion components of the sine wave response will affect measurement accuracy unless they are removed by narrow band filtering methods. If a narrow band filter is used, additional errors in the measurement due to filter accuracy are introduced.
A measurement system that can determine the frequency response of the SUT at many frequencies simultaneously can improve the speed of the measurement process. Such a measurement system consists of a stimulus generator connected to the SUT to apply an input stimulus thereto, and a two-channel spectrum analyzer with the first channel measuring the spectrum (S.sub.x) of the stimulus applied to the SUT and the second channel measuring the spectrum (S.sub.y) of the response of the SUT. In operation, the measurement system applies a stimulus of finite length from the stimulus generator to the SUT, detects the response of the SUT, measures the spectrums S.sub.x and S.sub.y, and calculates the transfer or impedance function using S.sub.x and S.sub.y. If the measurement system applies a stimulus of finite length to the input of the SUT, several methods (including a digital Fourier transform) can be used to compute a response at many discrete frequencies in a period of time equal to the time needed to measure a single frequency response to a sinusoidal stimulus applied to the SUT. The measurement time may thus be reduced in proportion to the number of spectral lines that may be computed in the time needed to measure the response of the SUT to a single sinusoidal stimulus. To implement this type of measurement requires the use of a broad band stimulus generator. With a broad band stimulus generator several methods may be used to determine the transfer function of the SUT from the input and output spectrums, S.sub.x and S.sub.y respectively. The ouptut spectrum, S.sub.y, can be divided by the input spectrum, S.sub.x, to form the transfer function of the SUT as shown in equation 1. ##EQU1## A better method is to compute the cross power spectrum between the stimulus and the response of the SUT as in equation 2. EQU G.sub.yx (f) = S.sub.y (f) S.sub.x *(f) (2)
The auto power spectrums of the stimulus and the response of the SUT are given in equations 3 and 4. EQU G.sub.xx = S.sub.x (f) S.sub.x *(f) (3) EQU G.sub.yy = S.sub.y (f) S.sub.Y *(f) (4)
The asterisk in each of equations 2, 3 and 4, and in each of the following equations where it appears, indicates the complex conjugate of the so designated function. From these quotations the transfer function H(f) may be computed as ##EQU2##
If averaged values for the cross and auto power spectrums are computed from an ensemble of sampled records then the transfer and coherence functions may be determined using a least squares estimation technique where ##EQU3## is the form of the least squares estimate of the transfer function and ##EQU4## is the coherence function for this estimate. The fraction of the SUT output power that is due to the SUT input power at a given frequency is represented by .gamma..sup.2 and has a value between 0.0 and 1.0. The bar above the functions in these equations indicates the average value of the so designated function. The least squares estimation technique of equations (6) and (7) is simply implemented using a digital processor capable of computing a digital Fourier transform that is well known in the art. These measurement techniques are more fully described in Roth, Peter R., "Effective Measurements Using Digital Signal Analysis," IEEE Spectrum, pp. 62-70, April 1971.
The most direct method of implementing the described measurement technique is to use a random noise stimulus which has a relatively flat spectrum over the band of frequencies being measured. A random stimulus has several important advantages in this measurement. It is easily made broad band in nature, delivering all frequencies in the band of interest. When used with the least squares technique of equations (6) and (7) the random stimulus is uncorrelated with noise and its own distortion products, and therefore yields an accurate estimation for H(f) in the presence of noise and nonlinearities.
The elimination of distortion components using a random stimulus is an important result of a measurement procedure which utilizes a random stimulus that is uncorrelated between records. In a spectrum of a random signal the relation between spectral lines is random from sample record to sample record when a collection of estimates is averaged together to yield a final result. Therefore, the distortion products that fall on any component of the response spectrum are uncorrelated from record to record. However, the response at each spectral line is deterministically related to the stimulus by the transfer function being measured. The result is that when the stimulus is uncorrelated with itself over a number of records, an ensemble record average of the cross spectrum between stimulus and response will reduce both noise and nonlinearity introduced by the SUT in proportion to the number of averages used.
The limitation on the use of random noise as a stimulus is the nonperiodicity of the continuous random noise. When any continuous signal is sampled for a finite period of time the spectrum of the resultant continuous signal of a finite record length is the spectrum of the continuous signal convolved with, or smeared by, the spectrum of the window function that is, or that is derived from, a sine function (i.e. sin x/x). The result is that each spectral line of the resultant frequency spectrum contains components from other frequencies. The effect on the measurement is that the transfer function is not the result of a measurement at a single frequency, but a measurement that is the weighted average given by the spectrum of the window function. This is analogous to the effect that would be observed using a continuous filter of finite bandwidth to observe the random signal. This phenomenon is called leakage in the literature and is more fully described in page 45 of Bergland, G. D., "A Guided Tour of the Fast Fourier Transform," IEEE Spectrum, Vol. 6, pp. 41-52, July 1969.
The leakage effect due to the continuous nature of random noise can be overcome by using a periodic broad band stimulus. Pseudo random sequences are one example of a periodic broad band stimulus, and these sequences are well known in the art. (One method for generating a pseudo random sequence is discussed in the "Operating and Service Manual" for the Hewlett-Packard Noise Generator Model 3722.) If the periodic stimulus has a period equal to the finite record length of the measured signal, leakage will be eliminated. In use, the periodic stimulus is applied to the SUT and allowed to repeat until the transient response from the SUT has decayed to a value small enough that the response of the SUT can be considered a periodic signal. The spectrum of the SUT response after the initial output transients have decayed to zero consists of a set of spectral lines spaced apart by .DELTA.f Hz, where .DELTA.f = 1/T and T is the period of the signal applied to the SUT. When a finite record length of a period T is used to compute the spectrum of the stimulus and the response of the SUT, each sinc function that is centered on each frequency of the applied stimulus is not affected by other spectral lines of the stimulus. This condition exists because the zeros of the sinc function spectrum of the sampling envelope are spaced .DELTA.f Hz apart, and all other lines of the spectrum fall on these zero points. A transfer function measured with a periodic stimulus will then measure the response of a system at each frequency with no effect from other frequencies.
The limitation of the periodic stimulus method is that different frequencies in the stimulus signal maintain a fixed relation to each other from measurement period to period. When nonlinearities are present in the SUT response, the distortion products at each frequency will maintain a fixed, correlated relation to the applied stimulus from record to record. The least squares technique will not eliminate the nonlinear components in the transfer function estimate, but will only discriminate against uncorrelated noise in the measurement. Because of the lower spectrum signal-to-noise ratio when broad band stimuli are used, distortion components in relatively linear systems can have a significant effect on measurement accuracy.